The answer to this is quite subtle and detailed and there is seldom agreement among even the medical fraternity about how much a child's mind is capable of as it develops. Zhro's answer is quite good in that it explains a lot about how the young child's mind learns, but it's controversial insofar as many gifted children learn and behave in a manner that we would normally associate with adults; many are specific cases of Asperger's Syndrome or more generally with the Autism spectrum. My goal is to provide that alternate view.
It is almost impossible to tell what parts of a child's development is a function of prior learning (or lack thereof) and what parts are a function of neurophysiological development (or lack thereof). But, in the case of some gifted children and certainly some (not all) cases of Asperger's, there's a clear case that learning and behaviour can cut both ways; sometimes the knowledge and the ability to learn is independent of physical age, other times, not so much.
To that end, an 8 YO child who's been that way for a millennium will have the depth of perspective one expects from someone that age and will be wise, knowledgeable and reasoned about many intellectual subjects. They will likely have a preferred style of art, music, food, etc. which is sophisticated insofar as it is drawn from a breadth of experience over time. Just how wise, knowledgeable and reasoned that person is will of course depend largely on how willing the person was through that millennium to explore, try new things, read, gain insights through thinking about those new experiences, etc. This is indeed no different to the rest of us. I've met many 'young' 70 YOs and a few 'old' 20 YOs in my time, and the difference usually comes down to experience and curiosity on the part of the individual.
That said, there will be some things our millennial (if you'll pardon the strange use of the term) won't have experienced because of the physical limitations of their body. Some of those should be evident, others not so much. Leadership for instance is something you wouldn't expect a millennial trapped in an 8 YO body to have vast amounts of experience in. This is largely due to the fact that many people would struggle to take a child with a high pitched voice seriously as a leader, regardless of their capability. The pains and difficulties of living in an older body, being a parent, public speaking; the list goes on. Our millennial isn't going to want to draw attention to his or her self for a start, so by virtue of that very fact many experiences the rest of us would take for granted (or at least aspire to) would be missed.
So your millenial is going to have a patchwork of development. Some areas will reveal a much deeper understanding of life, others will show him or her as almost naive.
Of course, this all assumes that the human mind can live that long and continue to store memories. We know for instance that human brains change as a person ages, and about the longest we've known one to function is around 150 years. True we can assume that our millennial has some advanced regenerative capabilities given that he or she has lived a thousand years already, but what we don't know is whether or not that regnerative capability would also impede the laying down of memories. If regneration also supports the brain, then it's entirely possible that our millennial would struggle to remember all but the most significant events in his or her life because the regeneration is wiping out memories to restore the brain. Conversely, if regeneration does NOT support the brain, our millennial may remember everything, but could be brain dead by 200 for all we know.
All in all, there are a quite a large number of variables in play here, but there's nothing to say that a perpetual 8 YO couldn't have a relatively normal development path (given their unique physiology), at least for a couple of centuries.